Wondering how you guys deal with this. Or is the inflatable material so think that you don't have to worry about sinking yourself?

While they could pose an issue its pretty unlikely. The boards are made from similar material as river running rafts (which are also used for fishing). And I have never heard of a raft being sunk by a hook

The main concern would be if you are fishing and using a gaff. I can see someone bringing in a nice yellow tail or halibut and gaffing it and putting it through the side of their board.